Question: Find the last two digits of the following sum: $$5! + 10! + 15! + \cdots + 100!$$
Solution: Since for all $n \ge 10$, $n!$ has (at least) two factors of 5, we know that $n!$ will end in two zeros.  Therefore, if $n\ge 10$, then $n!$ contributes nothing to the last two digits of the sum. So we need only compute $5! = 120$, and hence our answer is $\boxed{20}$.